Question:

A monopolist has an indirect demand function given by p(q), where p′(q) <0, The monopolist's marginal cost is constant and equal to c. In addition, the monopolist has to pay a per-unit tax of t on output it produces. The monopolist is a profit maximizer and chooses q.

(i) Write down an expression for profit and the necessary condition for an interior stationary point. Assume total revenue is concave. Check that the stationary pointis a maximum.

(ii) The condition in (i) implicitly defines q(c + t). Find an expression for q′(c + t)and sign it.

(iii) The government sets the tax to maximize tax revenue, T = t · q(c + t). Write down the necessary condition for an interior stationary point. Assume q′′(c+t) < 0 Check that the stationary point is a maximum.

(iv) The condition in (iii) implicitly defines t(c). Find an expression for t′(c) and sign it. Provide some intuition for this result

(i just need assistance with working and intuition, its question 1)

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ECOS2903 - Tutorial 4 - Answers
1.(i) Π(q) = p(q) · q − c · q − t · q
At a Stationary Point, Π′ (q) = p′ (q)q + p(q) − c − t = 0
Now, Π′′ (q) = p′′ (q)q + 2p′ (q). As total revenue is concave, Π′′ (q) ≤ 0 and stationary
point is a maximum.
(ii) q ′ (c + t) = 1
p′′ (q)q+2p′ (q) ≤ 0 because the denominator is Π′′ (q). (iii) At a stationary point T ′ (t) = q(c + t) + t · q ′ (c + t) = 0.
Now, T ′′ (t) = 2q ′ (·) + tq ′′ (·) ≤ 0 because q ′ () &lt; 0 and q ′′ () &lt; 0. Therefore SP is a
maximum.
′ ′′ q ()+t(c)q ()
′′
(iv) t′ (c) = − 2q
′ ()+t(c)q ′′ () ≤ 0. Note denominator is T (t). An increase in c decreases
q and so the marginal effect of t on T is smaller. As a result the optimal t decreases. 2. (i) AC(q) = C(q)
q (ii) AC ′ (q) = · (v ′ (q) − AC(q)) = 0. 1
q = F
q + v(q)
q . (iii) v ′ (q) = AC(q) or marginal cost equals average cost.
′
1
1
(iv) q ∗ (F ) = q∗ (F
) AC ′′ (q∗) &gt; 0.
(v) AC ∗ (F ) = F +v(q ∗ (F )) dAC ∗ (F )
.
dF
q ∗ (F ) = 1
q ∗ (F ) &gt;0 (vi) If F increases by ∆F , minimised average cost increases by approximately
that is, the increase in average fixed costs. ∆F
q∗ , 3. (i) p′ (q) &lt; 0 so strictly decreasing and one-to-one.
(ii) Inverse exists because one-to-one. Domain of q(p) is [0, p(0)]. q ′ (p) =
∫ p(0)
(iii) CS(p) = p
q(t)dt 1
p′ (q) &lt;0 (iv) Π(p) = CS(p) + (p − c) · q(p)
(v) For a stationary point, Π′ (p) = −q(p) + q(p) + (p − c) · q ′ (p) = 0. Or p = c.
Now Π′′ (p) = q ′ (p) + p · q ′′ (p), for a maximum this must be ≤ 0. Assume it is.
∫ p(0)
(vi) F = CS(c) = c
q(t)dt
(vii) The first order condition above states that maximise profit where price equals
marginal cost and extract all consumer surplus with the fixed fee. (Two part tariff).
You learnt this in first year. Other Questions
4. (i) Stationary points x = 1 and x = −1. Now f ′′ (x) = 6x. f ′′ (1) = 6 &gt; 0 so
function is convex around x = 1 and it is a local minimum. f ′′ (−1) = −6 &lt; 0 so
function is concave around x = −1 and it is a local maximum.
(ii) Stationary points x = 0 and x = −3. Now f ′′ (x) = ex (6x + 6x2 + x3 ). f ′′ (0) = 0
so x = 0 is an inflection pt. f ′′ (−3) &gt; 0 so function is convex around x = −3 and it
is a local minimum. 1 −3)
5. Stationary points at x = 1 and x = −1. f ′′ (x) = 2x(x
. Inflection points at
(1=x2 )3
√
√
x = 0, x = 3, x = − 3. At these three points the second derivative changes sign.
2 6. (a) L∗ = 160 and L∗∗ = 120. Check that maximum.
(b) No. Marginal product of labour goes through the maximum of average product.
You learnt this last year.
∫
1
7. (a) √1x dx = 2x 2 + C
∫
(b) 3e−2x dx = − 32 e−2x + C
8. (a) 8
(b) e − 1
e 9. F ′ (x) = x2 + 2 and G′ (x) = (x4 + 2) · 2x. 2

ECOS2903 - Tutorial 4
1. A monopolist has an indirect demand function given by p(q), where p′ (q) &lt;
0. The monopolist’s marginal cost is constant and equal to c. In addition, the
monopolist has to pay a per-unit tax of t on output it produces. The monopolist is
a profit maximizers and chooses q.
(i) Write down an expression for profit and the necessary condition for an interior
stationary point. Assume total revenue is concave. Check that the stationary point
is a maximum.
(ii) The condition in (i) implicitly defines q(c + t). Find an expression for q ′ (c + t)
and sign it.
(iii) The government sets the tax to maximize tax revenue, T = t · q(c + t). Write
down the necessary condition for an interior stationary point. Assume q ′′ (c + t) &lt; 0.
Check that the stationary point is a maximum.
(iv) The condition in (iii) implicitly defines t(c). Find an expression for t′ (c) and
sign it. Provide some intuition for this result.
2. Total cost is given by C(q) = F + v(q), where F is fixed cost and v(q) is variable
cost with v ′ (q) &gt; 0 and v ′′ (q) &gt; 0.
(i) Write down an expression for average cost, AC(q).
(ii) Write down the first-order condition for the q that minimises average cost. Assume the second-order condition for a minimum is satisfied.
(iii) Write this first order condition in terms of average and marginal cost.
′
(iv) This first-order condition implicitly defines q ∗ (F ). Find an expression for q ∗ (F )
and sign it.
(v) Substitute q ∗ (F ) into AC(q) to get AC ∗ (F ). Differentiate AC ∗ (F ) with respect
to F and simplify using (ii).
(vi) Sign the derivative in (v) and interpret.
3. Let the indirect demand function be given by p(q), q ∈ [0, q¯], where p(¯
q ) = 0 and
p′ (q) &lt; 0.
(i) Graph p(q) on a diagram with p on the vertical axis. Is p(q) one-to-one?
(ii) The direct demand function is given by q(p) and is the inverse of the indirect
demand function. What is its domain and what is the sign of q ′ (p)?
(iii) A monopolist can charge a fixed fee and a per-unit price. The fixed fee equals
consumer surplus at price p. Write an expression for consumer surplus as a function
of p. (hint: use the direct demand function)
(iv) Marginal cost is constant and equal to c. The monopolist’s total profit is the
sum of the fixed fee and the profit it earns by selling output at price p. Write down
an expression for the monopolist’s total profit as a function of p.
(v) Assume an interior solution. Find the price that maximises the monopolist’s
profit. (check that it is a maximum)
(vi) Write an expression for the fixed fee that maximises profit.
(vii) Discuss the economics. 1 Other Questions
4. Determine possible local extreme points for
(i) f (x) = x3 − 3x + 8
(ii) f (x) = x3 · ex
5. Decide whether f (x) = x
1+x2 is convex and determine possible inflection points. 6. A firm’s production function is Q(L) = 12L2 −
of workers, with L ∈ [0, 200]. 1 3
20 L , where denotes the number (a) What size of the work force, L∗ , maximises output? What size of the workforce,
L∗∗ , maximises output per worker?
(b) Note that Q′ (L∗∗ ) = Q(L∗∗ )
L∗∗ . Is this a coincidence? 7. Find the following integrals.
∫
(a) √1x dx
∫
(b) 3e−2x dx
8. Compute the area bounded by the graph of the function over the indicated
interval.
(a) f (x) = 3x2 over [0, 2].
(b) f (x) = 12 (ex + e−x ) over [−1, 1].
9. Put F (x) = ∫x
0 (t2 + 2)dt and G(x) = ∫ x2
0 2 (t2 + 2)dt. Find F ′ (x) and G′ (x).

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